ISSN 0806–2439 January 2009
A STOCHASTIC VARIANCE RATIO TEST
TO DISCRIMINATE BETWEEN TIME AND SPACE EFFECTS OF DISCREPANCY BETWEEN FILTRATIONS
PAUL C. KETTLER
ABSTRACT. This study provides three results of note. First, analysis of information can be abstracted from monetary values in special cases to variability of returns in general cases. Sec- ond, with an appropriately chosen model one can separate and calculate independently the influences on a stochastic process of having either more timely information, or better qual- ity information, or both. Third, one can bring methods of statistical inference to stochastic analysis. The paper develops a theory of applying variance ratio tests to problems of infer- ence, and by so doing, enables one to determine the separate influences of timely and superior information on results. Numerous examples illustrate the theory.
1. INTRODUCTION
The value of information as an abstract concept has been investigated by many. Among those directly relevant to mathematical inquiry are these (Øksendal 2005; Arriojas, Hu, Mo- hammed, and Pap 2007; Liu, Peleg, and Subrahmanyam 2007) and references therein.
The first of these papers delves into the values of three kinds of processes dependent on a nest of filtrations: a classical case with complete information, a partial observation case, and an insider case. The paper constructs a wealth process in each case, and imposes a log- arithmic utility function. The partial observation case employs an adapted process to the classical filtration, whereas the insider case requires a forward stochastic integral develop- ment for an anticipative process. Each process allows a value, which is the expectation of the logarithmic terminal wealth. The worth of the information lost or gained in the sub- and super-filtrations, therefore, is stated as the differences of these respective expectations.
The second of these papers investigates the value of information in the ‘constant absolute risk aversion’ (CARA) and ‘constant relative risk aversion’ (CRRA) settings, represented by power and exponential utility functions, respectively. The model presented is more general than that of the former paper, constructed to include consumption and allocation decisions, portfolio weights, stock beta values [of the Capital Asset Pricing Model (CAPM)], system- atic risk, and idiosyncratic uncertainty. The model is more intricate, therefore, but consistent with the norm of traditional financial economic studies, is less rigorous. The measure the- oretic probability methodology of the compared paper is more so, and thus its conclusions have greater force. Nonetheless, the latter paper forms interesting and believable conclu- sions on the value and impact of information, reinforced with an included empirical study.
One of these conclusions is that information is more valuable to the wealthy agent in the
Date: January 14, 2009.
2000Mathematics Subject Classification. Primary: 94A17, 40N30. Secondary: 91B70, 91B82.
1991Journal of Economic Literature Subject Classification. C13, C46.
Key words and phrases. Information; Stochastic process; Variance ratio.
The author wishes to thank Frank Proske and Olivier Menoukeu Pamen for valuable comments and suggestions.
1
CARA setting, but is insensitive to wealth in the CRRA setting. In both cases, information has significant impact on consumption and asset allocation.
The third of these papers studies a non-linear stochastic differential delay equation in a complete market setting, thus enabling a martingale measure equivalent to the assumed.
This study focuses on the delay of information specifically in the definition of the problem anticipating as to a Brownian motion adapted to its natural filtration.
Herein I present a variance ratio statistic, in analogy to the R2 statistic of linear regres- sion, but in reference to comparisons of the filtrations of stochastic process. Two versions appear, anR2T statistic for time-compared processes, and anR2S statistic for space-compared processes1. Each represents the fraction of variance of an included filtration ‘explained’ by the including one, and thus is a comparative measure between the two. In the real world of financial information, time decay is usually rapid, whereas spatial relations are more persis- tent, an observation offereden passant.
2. AVARIANCE RATIO STATISTIC
Assume filtered probability spaces
Ω,F,{Ft},P and
Ω,G,{Gt},P . Consider the fol- lowing constructions, whereσ2(·)is the variance assumed to exist a.s.P,∀t, 0 6t6T;u(·) is a utility function andJT is the terminal value of a performance functional, which could be wealth, and generally is dependent on several additional variables.
R2T:=1− σ2 E[u(JT)|Ft]
σ2 E[u(JT)|Fs], 06s6t6T (2.1)
R2S:=1− σ2 E[u(JT)|Gt]
σ2 E[u(JT)|Ft], Gt ⊇Ft
(2.2)
R2TS:=1− σ2 E[u(JT)|Gt]
σ2 E[u(JT)|Fs], 06s6t6T, Gt⊇Fs
(2.3)
R2
eTS:=1− σ2 E[u(JT)|Gs]
σ2 E[u(JT)|Ft], 06s6t6T, Gs⊇Ft
The mnemonic significance of the tilde over the ‘time’ index is that the roles ofsandtare reversed with respect to the formulation of Equation (2.3).
R2
TeS:=1− σ2 E[u(JT)|Ft]
σ2 E[u(JT)|Gs], 06s6t6T, Ft ⊇Gs
The mnemonic significance of the tilde over the ‘space’ index is that the roles ofF• andG•
are reversed with respect to the formulation of Equation (2.3).
Example 2.1(Brownian motion). Letu(Jt)be a Brownian motion, or equivalently, the logarithmic utility of a geometric Brownian motion. Let the lesser filtrationFsbe given by an information delay
1Observe that the subscript initials are set in roman, as they are mnemonic devices, not variables.
from timesto timet. Insofar a variance grows linearly with time, Equation(2.1)becomes R2T =1− T−t
T −s R2T = t−s
T−s
Example 2.2(Compound Poisson process). Letu(Jt)be a compound Poisson process, with in- tensityλandi.i.d. variables{Di = D}. The variance of the process isλtE[D2]. So, Equation(2.1) becomes
R2T=1− T −λtE[D2] T−λsE[D2] R2T= λtE[D2] −λsE[D2]
T −λsE[D2] R2T= (t−s)λE[D2]
T−λsE[D2] R2T= (t−s)λ
T/E[D2] −λs (2.4)
As a special case consider the Normal Inverse Gaussian distributionNIG(µ,α,β,δ), which has mean µ+δβ/γand varianceδα2/γ3, givenγ:=p
α2−β2. Let the mean be zero to produce a martingale.
ThenE[D2]equals the variance, and Equation(2.4)becomes R2T= (t−s)λ
(γ3/δα2)T −λs Example 2.3(Constant variance ratio process). Let
σ2 E[u(JT)|Gt]
=ρ2·σ2 E[u(JT)|Ft]
, 06ρ2 61 Then Equation(2.2)becomes
R2S=1−ρ2
Example 2.4(Linearly diminishing variance ratio process). Let σ2 E[u(JT)|Gt]
= (T −t)/T
ρ2·σ2 E[u(JT)|Ft]
, 06ρ2 61 Then Equation(2.2)becomes
R2S=1− T−t T ρ2 R2S= (1−ρ2) + t
Tρ2
Example 2.5(Brownian motion – constant variance ratio process).
R2TS=1− T −t
T −sρ2, 06ρ261
Example 2.6(Compound Poisson – linearly diminishing variance ratio process).
R2TS=1− T −λtE[D2]
T−λsE[D2]·T −t
T ρ2, 06ρ2 61 As a special case consider the Normal Inverse Gaussian distributionNIG(µ,α,β,δ).
R2TS=1− T −λt·δα2/γ3
T−λs·δα2/γ3 ·T−t
T ρ2, 06ρ2 61
Example 2.7(Brownian motion – constant variance ratio process, time reversed).
R2
TSe =1− T−s
T −tρ2, 06ρ26 T−t T −s
Example 2.8 (Compound Poisson – linearly diminishing variance ratio process, space re- versed).
R2
TeS =1− T −λtE[D2]
T −λsE[D2]·T−t
T ρ2 , T−λtE[D2]
T−λsE[D2]·T−t
T 6ρ2<∞ As a special case consider the Normal Inverse Gaussian distributionNIG(µ,α,β,δ).
R2
TeS =1− T −λt·δα2/γ3
T −λs·δα2/γ3 ·T−t
T ρ2 , T−λt·δα2/γ3
T−λs·δα2/γ3 ·T −t
T 6ρ2<∞ 3. MODEL-DIRECTEDF-RATIO TESTS FOR EQUALITY OF FILTRATIONS
If one chooses a model, for instance one of those in the examples of Section 2, then an examination of data gathered pursuant to the model provides parameter estimates by one of several methods. Maximum likelihood [ML] comes immediately to mind, as do other procedures, such as ordinary least squares [OLS]. With parameters selected from one data set, viewed as a control set, then one may take another, independent, sample, generating statistics from the model, in particular the variousR2•statistics described.
A simple F-ratio test for equality of variance is not useful in these cases, for the tacit as- sumption is that variances are not equal, except for the [uninteresting] cases of equality of filtrations. However, that said, a reduced F-ratio test is feasible, constructed by compar- ing the variance with respect to the smaller filtration to the variance of the larger filtration reduced by the factor 1−R2•. In this regard, recast Equations (2.1) and (2.2) as follows.
σ2 E[u(JT)|Ft]
(1−R2T)σ2 E[u(JT)|Fs] ∝1, 06s6t6T σ2 E[u(JT)|Gt]
(1−R2S)σ2 E[u(JT)|Ft] ∝1, 06t6T
Restating the examples of Section 2 gives this new set, which serve as direct models for applying the F-ratio test for fitness.
Example 3.1(Brownian motion).
t−s R2T
T−s ∝1
Example 3.2(Compound Poisson process).
(t−s)λ R2T
T/E[D2] −λs ∝1 And for the special case of theNIGprocess —
(t−s)λ R2T
(γ3/δα2)T −λs ∝1 Example 3.3(Constant variance ratio process).
1−ρ2
R2S ∝1, 06ρ261 Example 3.4(Linearly diminishing variance ratio process).
(1−ρ2) + t Tρ2
R2S ∝1, 06ρ2 61 Example 3.5(Brownian motion – constant variance ratio process).
T −t
T −sρ2∝1, 06ρ26 T −s T−t
Example 3.6(Compound Poisson – linearly diminishing variance ratio process).
T −λtE[D2]
T−λsE[D2]·T −t
T ρ2 ∝1, T−λsE[D2] T−λtE[D2]· T
T −t 6ρ2<∞ (3.1)
As a special case consider the Normal Inverse Gaussian distributionNIG(µ,α,β,δ).
T−λt·δα2/γ3
T−λs·δα2/γ3 ·T −t
T ρ2 ∝1, T−λs·δα2/γ3 T−λt·δα2/γ3 · T
T −t 6ρ2<∞ Example 3.7(Brownian motion – constant variance ratio process, time reversed).
T −s
T −tρ2∝1, 06ρ26 T−t T −s
Example 3.8 (Compound Poisson – linearly diminishing variance ratio process, space re- versed).
T −λtE[D2]
T−λsE[D2]·T −t
T ρ2 ∝1, T −λtE[D2]
T−λsE[D2]·T −t
T 6ρ2<∞ As a special case consider the Normal Inverse Gaussian distributionNIG(µ,α,β,δ).
T−λt·δα2/γ3
T −λs·δα2/γ3 ·T −t
T ρ2 ∝1, T −λt·δα2/γ3
T−λs·δα2/γ3 ·T −t
T 6ρ2 <∞
4. INSIDER TRADES
The way is clear now to specify tests for the purpose of deciding whether or not certain trading patterns reveal the use of inside information. The superior information can come in the form of a time advantage or a space advantage, or both. In the former instance infor- mation is acquired by the insider before it is available to the general trading public; in the latter case the information is of premium quality. The usual case in actual market circum- stances is that the incremental information is both better and earlier than that available to the lesser informed populace. In practice, the time advantage is evanescent, whereas the space advantage is persistent.
Take, for illustration, Equation (3.1) of Example 3.6, repeated below, the variational ex- pression for the compound Poisson – linearly diminishing variance ratio process. Assume that all quantities are known, except forsandρ2. Determining these parameters establishes the effects, respectively, of the time and space discrepancies in information. Variances of samples of returns from trading patterns from each realm — superior and inferior informa- tion — serve to define implicitly the relationship between the parameters. Let, therefore, (4.1) ϕ(s,ρ2):= T−λt·δα2/γ3
T −λs·δα2/γ3 ·T−t
T ρ2∝1, T −λs·δα2/γ3 T −λt·δα2/γ3 · T
T−t 6ρ2<∞ The first factor serves to reduce the expected variation in the returns of the lesser informed trader to that of the insidervis à visthe time differential, whereas the second factor serves the same purposevis à visthe space differential. The question arises, “How does one choose a pair?” Insofar as an increased discrepancy of information can be explained by lower values in eithersorρ2, it is correct to assume an inverse relationship between them. The examples of this section including both variables, in particular Equation (4.1), confirm this view.
One choice is to take a Bayesian ‘diffuse prior’ approach, and set each factor ofϕ(s,ρ2)to 1. For this discrete case the choice also provides maximal entropy. Then,
s=t ρ2= T
T−t
Thus, in this case the entire insider advantage is in the quality of information, not in the time- liness of its arrival. In the complementary case,s = 0, providing maximal time advantage, leaving
ρ2= T2
(T −λt·δα2/γ3)(T−t)
Another choice involves evaluating the incremental information to an insider acquired from each source — earlier acquisition (smaller value of s) or improved quality (smaller value ofρ2), while assuming that the marginal costs for equal value are the same in equilib- rium. The idea is to minimize an expression of the form
as+bρ2 subject to a constraint such as
sρ2=k, (4.2)
wherea > 0,b > 0, and k > 0 are constant coefficients. This optimization problem lends itself to a Lagrange multiplier approach, wherein the Lagrangian is
Λ(s,ρ2;λ) =as+bρ2−λ(sρ2−k) A solution is provided by solving the system
∂Λ
∂s =a−λρ2=0
∂Λ
∂ρ2 =b−λs =0,
along with the equation of constraint, Equation (4.2). This solution is (s,ρ2) =
rbk a ,
rak b
!
λ= rab
k
An interpretation of the Lagrange multiplierλis that of the marginal cost of the combined time and space information at the exchange rate implied by Equation (4.2) at the optimum.
5. SAMPLE AND TEST PROCEDURES
Consider any of the generic R2 statistics of the previous section. The value 1− R2 is a variance ratio, dividing in each instance the variance for the lesser filtration by the variance for the greater filtration. If this value is estimated by sampling respectively(n1,n2) points from the joint distribution, then Fisher’s F-statistic is the ratio of variances calculated for (ν1,ν2) = (n1−1,n2−1)[numerator and denominator] degrees of freedom. For example, if (ν1,ν2) = (10, 20), then at the 5% significance level one has
F.05(ν1,ν2) =F.05(10, 20) =2.348
Inverting this relationship implies that for anF-ratio of 2.348, thep-value is .05. For anF- ratio of 3.000 thep-value drops to .017510 with the same degrees of freedom. Tables and on- line calculators are readily available for such calculations. The values herein were gleaned from (Soper 2009).
It is well known that the ordinaryF-ratio test, as described, becomes unreliable as the dis- tributions depart from normal. However, one can take this matter into account by devising p-value intervals, as done with the Durbin – Watson test,e.g., for auto-correlation of regres- sion residuals. This development is outside the scope of the present tract, but is suggested for further research, including the preparation of appropriate tables and on-line calculators.
6. CONCLUSIONS
This study provides three results of note.
(1) Analysis of information can be abstracted from monetary values in special cases to variability of returns in general cases.
(2) With an appropriately chosen model one can separate and calculate independently the influences on a stochastic process of having either
(a) more timely information, or
(b) better quality information, or both.
(3) One can bring methods of statistical inference to stochastic analysis.
REFERENCES
Arriojas, M., Y. Hu, S.-E. Mohammed, and G. Pap (2007). A delayed Black and Scholes formula I.J. Stoch. Anal. Appl. 25(2), 471–492.
Liu, J., E. Peleg, and A. Subrahmanyam (2007, Oct.). Information, expected utility, and portfolio choice. Available at http://www.s1ubra.org/lps2.pdf.
Øksendal, B. K. (2005, Dec.). The value of information in stochastic control and finance.
Aust. Econ. Pap. 44(4), 352–364.
Soper, D. S. (2009, January). p-value calculator for the Fisher F-test. Internet publication.
http://www.danielsoper.com/statcalc/calc07.aspx.
(Paul C. Kettler)
CENTRE OFMATHEMATICS FORAPPLICATIONS
DEPARTMENT OFMATHEMATICS
UNIVERSITY OFOSLO
P.O. BOX1053, BLINDERN
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E-mail address:paulck@math.uio.no URL:http://www.math.uio.no/∼paulck/
